Spontaneous generating edges?
That's the result of Triangulation of NGons in the game engine, it doesn't effect the original geometry.

in game mode

Back out of Game Engine, the geometry is not changed. If it bothers you the solution is to make the geometry without NGons in the first place. This means cutting up the NGon into triangles or quadrangles. A fast way to convert an NGon to the elementary polygon types is:
Ngon to Tris

in edit mode

select the ngon

press Ctrl+t (this creates Triangles)
should generate something like this



Tris to Quads

With a set of Triangles selected press Alt + j , and blender will try to make quads from the geometry.

Sometimes it isn't possible to convert all selected Triangles into a set of Quads. invariably there will be times when the algorithm leaves a few Triangles, even though geometrically it would be possible to neatly Quadify all selected geometry. This is not a bug (really, we could argue about this), but rather a limitation in the algorithm.
this might produce geometry like this:

The algorithms that do this don't have the benefit of our eyes and brains, to make the best decisions. Sometimes you have to do the tessellation yourself